Blossoming

In the first case L(t) preserves linear combinations; in the second case L(t) preserves affine combinations. But if P(t) ¼ antn þ þ a0 is a polynomial of degree n > 1, then

P(msþ lt) 6¼ mP(s)þ lP(t), P((1 l)sþ lt) 6¼ (1 l)P(s)þ lP(t):

Thus arbitrary polynomials preserve neither linear nor affine combinations. The key idea behind blossoming is to replace a complicated polynomial function P(t) in one variable by a simple polynomial function p u1, . . . , unð Þ in many variables that is either linear or affine in each variable. The function p u1, . . . , unð Þ is called the blossom of P(t), and converting from P(t) to p u1, . . . , unð Þ is called blossoming.